Order quantity flexibility as a form of customer service in a supply chain contract model

Abstract

This paper analyzes the Quantity-Flexibility (QF) contract, under which the buyer provides to the supplier information about expected future orders for the predetermined horizon and the supplier, in return, provides the buyer with the flexibility to adjust future orders later. Under this scheme, the flexibility profile of the contract can be perceived by the buyer as a form of customer service, by which the supplier commits to fulfil the buyer’s maximum likely order at the cost of the supplier’s inventory risk. The simulation results show that the benefit of the contract to either party clearly depends on the flexibility profile of the contract. For the supplier, there exists a trade-off between customer service level and inventory risk associated with the flexibility profile. However, for the buyer, the flexibility demonstrates a principle of diminishing returns, which is contrary to the general notion that buyers always prefer more order quantity flexibility. In fact, greater flexibility does not always translate into better customer service to buyers.

Appendix

In the new models, both the retailer and the manufacture in Model 1 of Fig. 2 now employ an optimal ordering model based on an adaptive OUT inventory policy, respectively. In Model 2, the retailer now employs the adaptive OUT inventory policy and the EWMA, and enters into a QF contract with the manufacturer. Therefore, both the retailer and the manufacturer apply the QF scheme to estimate their current orders, and only the retailer applies the scheme to estimate its future orders in the chosen rolling-horizons. More detailed equations for the new models are provided below.

Assume that the values of τ, φ and $\sigma_{\varepsilon }$ of an AR(1) process are known to the retailer. Let OUT^R (t) denote an order-up-to level of the retailer in period t and m^R (t) and v^R (t) denote the conditional expectation and the conditional variance of the total demand over the lead time, respectively. Then the optimal order-up-to level, OUT^R* (t), that minimizes the total expected inventory holding and backorder costs can be found as follows (Lee et al. 1997, 2000):

$$ {\text{OUT}}^{\text{R*}} ( {\text{t)}} = {\text{m}}^{\text{R}} ( {\text{t)}} + {\text{K}}^{\text{R}} \sigma_{\varepsilon } \sqrt {{\text{v}}^{\text{R}} ( {\text{t)}}} $$

where ${\text{m}}^{{\text{R}}} ({\text{t)}} = {\text{Exp}}\left[ {\sum\nolimits_{{{\text{i = 1}}}}^{{{\text{od}} + {\text{sd}}}} {{\text{D}}({\text{t + i}}){\text{|D}}({\text{t}})} } \right]$, ${\text{v}}^{{\text{R}}} ({\text{t)}} = {\text{Var}}\left[ {\sum\nolimits_{{{\text{i = 1}}}}^{{{\text{od}} + {\text{sd}}}} {{\text{D}}({\text{t + i}}){\text{|D}}({\text{t}})} } \right]$, ${\text{K}}^{\text{R}} = \Upphi^{ - 1} \left[ {{\frac{{{\text{Backorder\_cost}}}}{{{\text{Backorder\_cost }} + {\text{ Holding\_cost}}}}}} \right]$ (Φ: Standard Normal dist. func.)

Then, according to Lee et al. (2000) and Agrawal et al. (2009), the retailer replenishes the demand during period t plus the change being made in the order-up-to levels as follows:

$$ {\text{O}}_{0}^{{\text{R}}} ({\text{t)}} = {\text{D}}({\text{t}}) + \left[ {{\text{OUT}}^{{{\text{R*}}}} ({\text{t}}) - {\text{OUT}}^{{{\text{R*}}}} ({\text{t}} - 1)} \right] $$

and the resulting optimal order quantity of the retailer is given by

$$ {\text{O}}_{ 0}^{\text{R}} \left( {\text{t}} \right) = {\text{D}}\left( {\text{t}} \right) + {\frac{{\phi (1 - \phi^{{{\text{od}} + {\text{sd}}}} )}}{1 - \phi }}\left[ {{\text{D}}\left( {\text{t}} \right) - {\text{D}}\left( {{\text{t}} - 1} \right)} \right] $$

where φ is a correlation coefficient in the AR(1) process

Likewise, with the following similar notations for the manufacturer, OUT^M(t), m^M(t) and v^M(t), the optimal order-up-to level, OUT^M*(t), that minimizes the total expected inventory holding and penalty costs can be found as follows (Lee et al. 2000):

$$ {\text{OUT}}^{\text{M*}} ( {\text{t)}} = {\text{m}}^{\text{M}} ( {\text{t)}} + {\text{K}}^{\text{M}} \sigma_{\varepsilon } \sqrt {{\text{v}}^{\text{M}} ( {\text{t)}}} $$

where ${\text{K}}^{\text{M}} = \Upphi^{ - 1} \left[ {\frac{{{\text{Penalty\_cost}}}}{{{\text{Penalty\_cost }} + {\text{ Holding\_cost}}}}} \right]$

Then the manufacturer replenishes the order from the retailer plus the change being made in the order-up-to levels as follows (Lee et al. 2000; Agrawal et al. 2009):

$$ {\text{O}}_{ 0}^{\text{M}} ( {\text{t)}} = {\text{O}}_{ 0}^{\text{R}} ( {\text{t}} - {\text{od)}} + \left[ {{\text{OUT}}^{\text{M*}} ( {\text{t)}} - {\text{OUT}}^{\text{M*}} ( {\text{t}} - 1 )} \right] $$

and the resulting optimal order quantity of the manufacturer is given by

$$ {\text{O}}_{ 0}^{\text{M}} ( {\text{t)}} = {\text{O}}_{ 0}^{\text{R}} ({\text{t}} - {\text{od}}) + {\frac{{\phi (1 - \phi^{\text{pd}} )}}{1 - \phi }}\left[ {{\text{O}}_{ 0}^{\text{R}} ( {\text{t}} - {\text{od)}} - {\text{O}}_{ 0}^{\text{R}} ( {\text{t}} - {\text{od}} - 1 )} \right] $$

Now, in Model 2, the retailer uses the EWMA in Eq. 25 for forecasting demands of the chosen rolling horizons.

Simulations are conducted with the same settings specified in the Sect. 4.1, and the resulting outcomes are shown in Figs. 7, 8, 9 and Table 3.

Fig. 7

Order variance ratios of the retailer (R) and the manufacturer (M) versus flexibility rate

Fig. 8

Average fill rate versus flexibility rate

Fig. 9

Cost savings of the retailer (R) and the manufacturer (M) versus flexibility rate

Table 3 Total costs of the retailer (R) and the manufacturer (M)

Figure 7 shows the Bullwhip effects in the new models. We can see that they are overall mitigated compared to the outcomes in Fig. 3, although the order variance ratios of the manufacturer still vary widely with the flexibility rate: the higher the flexibility rate, the higher the order variance ratio of the manufacturer. However, one thing to note is that the dashed line (no QF contract case) in Fig. 7 now exists right below the case of the flexibility rate 25%, while that in Fig. 3 exists below the case of the flexibility rate 100%. This implies that the manufacturer in the new models may request much smaller flexibility rate when entering into the QF contract with the retailer.

Figure 8 shows the average fill rates in the new models. The main difference from the outcomes in Fig. 6 is that no QF contract case provides a quite high fill rate as it is. Therefore, from the perspective of the customer service level only, the retailer might not be interested in entering into the QF contract. However, the retailer eventually needs to make the decision after taking into account the cost performances as well, as will be explained below.

Both Table 3 and Fig. 9 show cost performances of the retailer and the manufacturer in the new models. They show that both the retailer’s and the manufacturer’s cost performances have improved with the new ordering model, in both the QF contract and no QF contract cases, compared to the outcomes in Table 1 and Fig. 5. Another thing to note is that the resulting feasible contract range is now from 25% to 30%, which is smaller than that in Table 1 and Fig. 5. This is because the lower flexibility bound of the feasible contract range for the retailer has increased and the upper one of that for the manufacturer has decreased. Other than the abovementioned differences, major findings and implications remain the same.